# IA-automorphism group of nilpotent group equals stability group of lower central series

## Statement

### For an individual automorphism

Suppose $G$ is a nilpotent group. Then, the following are equivalent for an automorphism $\sigma$ of $G$:

1. $\sigma$ is an IA-automorphism of $G$, i.e., it induces the identity map on the abelianization of $G$.
2. $\sigma$ is a stability automorphism for the lower central series of $G$.

Note that $G$ being nilpotent is important only in so far as it guarantees that the lower central series reaches the trivial subgroup. A slight variant of the statement would be true for non-nilpotent groups, but we wouldn't use the jargon of stability automorphism.

### For subgroups of the automorphism group

Suppose $G$ is a nilpotent group. The following subgroups of the automorphism group of $G$ are equal:

1. The subgroup of IA-automorphisms of $G$, i.e., automorphisms that induce the identity map on the abelianization of $G$.
2. The stability group (i.e., the group of stability automorphisms) for the lower central series of $G$.

## Proof

### Background fact on iterated commutator mapping

The proof basically follows from the fact that for any positive integer $n$, the left-normed iterated commutator gives a surjective $n$-linear map from the abelianization of $G$ to the quotient group $\gamma_n(G)/\gamma_{n+1}(G)$ between successive members of the lower central series. Here, $\gamma_n(G)$ is the $n^{th}$ member of the lower central series of $G$. Explicitly, it is a $n$-linear map of abelian groups:

$G/G' \times G/G' \times \dots G/G' \to \gamma_n(G)/\gamma_{n+1}(G)$

This mapping is canonical, hence covariant with automorphisms.

### (1) implies (2)

Given: An IA-automorphism $\sigma$ of a nilpotent group $G$.

To prove: $\sigma$ induces the identity map on each of the successive quotients of the lower central series, i.e., for every $n$, $\sigma$ induces the identity map on each of the groups of the form $\gamma_n(G)/\gamma_{n+1}(G)$ where $\gamma_n(G)$ is the $n^{th}$ member of the lower central series of $G$.

Proof: The covariance of the iterated commutator mapping with respect to $\sigma$, along with its surjectivity to $\gamma_n(G)/\gamma_{n+1}(G)$, guarantees that since $\sigma$ fixes $G/G'$ pointwise, it also fixes the image $\gamma_n(G)/\gamma_{n+1}(G)$ pointwise.

### (2) implies (1)

Given: A nilpotent group $G$. An automorphism $\sigma$ induces the identity map on each of the successive quotients of the lower central series, i.e., for every $n$, $\sigma$ induces the identity map on each of the groups of the form $\gamma_n(G)/\gamma_{n+1}(G)$ where $\gamma_n(G)$ is the $n^{th}$ member of the lower central series of $G$.

To prove: $\sigma$ is an IA-automorphism of $G$.

Proof: This follows directly from setting $n = 1$.